Problem: How many two-digit prime numbers have a units digit of 7?
Answer: The tens digit cannot be 2, 5, or 8 because otherwise, the sum of the digits, and thus the number itself, is divisible by 3. The remaining possibilities are 17, 37, 47, 67, 77, and 97. Only $77=7\cdot11$ is composite, so there are $\boxed{5}$ two-digit prime numbers that have a units digit of 7.